The Nonarchimedean Scottish Book

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The original Scottish Book was a compilation of problems, primarily concerning functional analysis, assembled at the Scottish Cafe in Lvov during the 1930s and 1940s. This web site is an analogous compilation of problems concerning nonarchimedean functional analysis and related topics, including analytic geometry (especially adic spaces) and perfectoid rings, fields, and spaces.

An ideal entry in this list includes the name of the proposer, the date of the initial post, a detailed statement of the problem, and attributed comments (including references as appropriate; these can be listed at the bottom of the page).

For security reasons, editing this wiki requires logging into the server. I have created a communal login credential for editing this page; please contact me for the details. Alternatively, I am wiling to accept entries by email and post them manually. (If someone with more free time than me wants to reimplementing this using better technology, I would support this.)

Contents

Problem 1

Proposed by Kiran S. Kedlaya, 17 December 2015.

Statement: Let K be a nonarchimedean commutative Banach ring whose underlying ring is a field. Suppose in addition that K is uniform, i.e., its norm is equivalent to a power-multiplicative norm. Is K necessarily a nonarchimedean field, that is, is the topology on K defined by some multiplicative norm?

Proposer's comment: This question acquires a negative answer if the uniform condition is omitted. Details available upon request.

Problem 2

Proposed by Kiran S. Kedlaya, 17 December 2015.

Statement: Let (A,A^+) be a Huber pair which is Tate (i.e., A contains a topologically nilpotent unit). Suppose that Spa(A,A^+) is a perfectoid space (for some prime p). Is A necessarily a perfectoid algebra?

Comment (Kedlaya, 18 Dec 2015): This is [Sch2, Conjecture 2.16], but a counterexample is known [BV, Proposition 13]. However, if one further assumes that A is uniform (or even stably uniform), the problem is open in general; see [KL2, Remark 3.6.27]. If A is of characteristic p and sheafy, then the statement holds; see [BV, Corollary 10], [KL1, Proposition 3.1.16].

Problem 3

Proposed by David Hansen, 17 December 2015.

Statement: Let A be a perfectoid Tate ring with an action of a finite group G. Is the fixed subring A^G perfectoid?

Proposer's comment: This is true if A is of characteristic p, by an easy argument. This is also true if A is a Q_p-algebra, by a more subtle argument of Kedlaya.

Problem 4

Proposed by David Hansen, 18 December 2015.

Statement: Let A be a sheafy Tate ring, and suppose some Zariski-open subset of some Spa(A,A^+) is a perfectoid space. Is A perfectoid?

Proposer's comment: This is a stronger version of Problem 2.

Problem 5

Proposed by David Hansen, 18 December 2015.

Statement: Let A be a stably uniform Tate ring over Q_p, and let R^+ \subset R be a perfectoid Tate ring in characteristic p. Consider the Tate ring (W(R^+) \widehat{\otimes} A^\circ)[1/p], where the completion is taken for the p-adic topology. Is this ring stably uniform?

Proposer's comment: The answer is yes when A is finite etale over some Q_p<X_1,...,X_n>.

Problem 6

Proposed by David Hansen, 18 December 2015.

Statement: Call a Tate ring A "sousperfectoid" if there exists a perfectoid Tate ring B and a continuous A-algebra map A-->B which admits a continuous A-Banach module splitting. This class includes perfectoid Tate rings, and any sousperfectoid ring is stably uniform and hence sheafy. If R is sousperfectoid, then any rational localization of R is sousperfectoid, and so are R<X>, R<X^1/p^infty> and R' for any finite etale R'/R.

Is there an example of a stably uniform Tate ring which is not sousperfectoid?

Comment (Kedlaya, 18 Dec 15): For A to be sousperfectoid, a necessary condition is that A be seminormal in the sense of Greco-Traverso and Swan: for any y, z in A for which y^3 = z^2, there is a unique x in A with x^2 = y, x^3 = z. (This will be shown in [KL2].)

Comment (Kedlaya, 18 Dec 15): Let X = Spa(A,A^+) be a classical affinoid algebra over a p-adic field. Let f: Y -> X be a resolution of singularities. If A is seminormal and R^1 f_* O_Y = 0, then I believe I can prove that A is sousperfectoid. For example, this holds if X has rational singularities because then all of the R^i f_* O_Y vanish.

Problem 7

Proposed by Kiran S. Kedlaya, 18 December 2015.

Statement: Let (A,A^+) be a sheafy uniform (Tate) Huber pair. Is (A,A^+) necessarily stably uniform?

Proposer's comment: This question is taken from [KL1, Remark 2.8.11].

Problem 8

Proposed by Kiran S. Kedlaya, 18 December 2015.

Statement: Let (A,A^+) be a Huber pair. Is the property of (A,A^+) being sheafy, or stably uniform, independent of the choice of A^+?

Problem 9

Proposed by Kiran S. Kedlaya, 18 December 2015.

Statement: Let (A,A^+) be a stably uniform Huber pair. Is it true that for any finite etale morphism (A,A^+) -> (B,B^+), the pair (B,B^+) is again stably uniform?

Proposer's comment: This is true if A is sousperfectoid (see problem 6), as then B is as well.

References

[BV] K. Buzzard and A. Verberkmoes, Stably uniform affinoids are sheafy, arXiv:1404.7020v2 (2015).

[KL1] K.S. Kedlaya and R. Liu, Relative p-adic Hodge theory: Foundations, Asterisque 371 (2015).

[KL2] K.S. Kedlaya and R. Liu, Relative p-adic Hodge theory, II: Imperfect period rings, in preparation.

[Sch1] P. Scholze, Perfectoid spaces, Publ. Math. IHES.

[Sch2] P. Scholze, Perfectoid spaces: a survey, in Current Developments in Mathematics, 2012, International Press, Boston, 2013.

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